A note on persistence and extinction of a randomized ...

Applied Mathematics and Computation 246 (2014) 599–607

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A note on persistence and extinction of a randomized food-limited logistic population model Dianli Zhao ⇑, Sanling Yuan College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

a r t i c l e

i n f o

Keywords: Stochastic logistic model Food-limited assumption Global stability Lyapunov functional

a b s t r a c t This paper addresses the issue of the asymptotic behavior for a non-autonomous randomized food-limited logistic population model. Several sufficient conditions are formulated and proved for p-moment persistence and extinction of the population, as well as in sense of almost sure. Results show that food-limited assumption has an influence on the convergence rate of the solution to the equilibria for the deterministic and stochastic model. Some previously known results are improved. Numerical simulations are provided to support the results. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction A logistic differential equation with stochastic perturbation, or shortly, a stochastic logistic equation, is a system of the form

  NðtÞ dNðtÞ ¼ NðtÞ 1  ½rdt þ rdBðtÞ: K

ð1Þ

It has been studied by many authors as population growth model of a single species (see [1–3]). NðtÞ denotes the population density at time t; r is called the intrinsic rate of growth and K is the carrying capacity assumed to be positive. BðtÞ is a standard Brown motion representing the effects induced by the environmental noise on the natural growth. r is the intensity of the noise. Considering the influence of environment, Golec and Sathananthan [4] assume that the coefficients of the model are time-varying and they investigate the stability of non-autonomous randomized population model

  NðtÞ dNðtÞ ¼ NðtÞ 1  ½rðt Þdt þ rðtÞdBðtÞ; K

ð2Þ

where r ðt Þ and rðt Þ are both continuous functions on ½0; þ1Þ. By assuming that rðtÞ  r; rðtÞ  r and 0 < N 0 < K, known results presented in [1] and [4] are as follows. (A1) If r <  12 r2 , then limt!1 NðtÞ ¼ K, a.s. (almost surely); (A2) If r > 12 r2 , then limt!1 NðtÞ ¼ 0, a.s. ⇑ Corresponding author. E-mail address: [email protected] (D. Zhao). http://dx.doi.org/10.1016/j.amc.2014.08.070 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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In [5,6], the Allee effect is considered on solution of the stochastic population model.   . It is violated for nearly all populations, e.g. for a food-limited The determined version of (1) is like dNðtÞ ¼ rNðtÞ 1  NðtÞ dt K population. An experiment analysis done by Smith [7] shows the classical logistic model does not fit experimental data very well, and suggested a modification of the logistic equation:

dNðtÞ dt

KNðtÞ of increase with unlimited food ¼ rNðtÞ KþCNðtÞ where C ¼ rate rate is a of replacement of mass in thesaturation

nonnegative constant denoting the delayed effects of the food-limited on the growth of the population. From then on, a lot of authors consider the effect of the food-limited on modified logistic model. The existence, uniqueness, and global stability of positive solutions, periodic solutions and Hopf bifurcation of the deterministic food-limited logistic equation have been investigated (see, for example, [8–12]). In [13], Jiang et al. study a randomized logistic equations with food-limited effect of the form

dNðtÞ ¼ NðtÞ

K  NðtÞ ½rdt þ rdBðtÞ; K þ CNðtÞ

ð3Þ

where C; r; r 2 ½0; 1Þ are both constants. They get the following result. Theorem A. Let NðtÞ be a continuous positive solution to Eq. (3) for any initial value Nð0Þ ¼ N 0 with 0 < N 0 < K. If r > r2 , then h i limt!1 E ðNðtÞ  K Þ2 ¼ 0. Here, EðXÞ denotes the expectation of X. In the sequel, L1 ½0; 1Þ denotes the family of all the continuous and integrable functions on ½0; þ1Þ. Z þ ¼ f1; 2; 3; . . .g. h i By comparing the above results, we conjecture that limt!1 E ðNðtÞ  K Þ2 ¼ 0 may be deduced from the condition r > 12 r2 , and the food-limited assumption may affect the condition for stability of the equilibrium. Almost sure persistence and extinction is also important problem for study of the population modeled by (3), but no results are found by the authors up to now. In reality, due to the influence of environmental changes, such as changes in weather, habitat destruction and exploitation, the expanding food surplus and other factors, the growth rate and the noise intensity in model (3) may be time-independent (see, for example, [4,14–16]). An interesting problem is whether the time-varying coefficients assumption can be useful for obtaining some results different from that in [13]. Motivated by the above, in this paper, the authors will mainly focus on finding the conditions for the persistence and extinction of population modeled by the non-autonomous randomized food-limited population equation

dNðtÞ ¼ NðtÞ

K  NðtÞ ½rðt Þdt þ rðtÞdBðtÞ; K þ CNðtÞ

ð4Þ

where rðtÞ and rðt Þ are both bounded and continuous functions on ½0; þ1Þ. Note that (4) has two equilibria: 0 and K. 1.1. Deterministic food-limited logistic population model Before statement of the main results, we list the properties of the deterministic logistic model with food-limited assumption for comparison later. Consider

dNðtÞ K  NðtÞ ¼ rðtÞNðtÞ : dt K þ CNðtÞ

ð5Þ

It’s clear that 0 and K are also the equilibria of the deterministic logistic model (5). The following is mainly about the conditions for convergence of the solution to the equilibrium. Theorem 1.1. Let NðtÞ be a solution of Eq. (5) with Nð0Þ ¼ N 0 > 0. If limt!1

Rt

0

rðsÞds ¼ 1, then limt!1 NðtÞ ¼ K.

Proof. Let NðtÞ be a solution of (5). It is obvious if there is t0 P 0 such that Nðt 0 Þ ¼ K, then NðtÞ ¼ K for all t > t 0 . Note that NðtÞ < K implies that dNðtÞ P 0, and NðtÞ > K implies that dNðtÞ 6 0 for N 0 > 0, if NðtÞ – K for all t > 0, the solution of (5) can be dt dt presented as

NðtÞ jK  NðtÞj1þC

¼

Therefore, if limt!1

N0 jK  N 0 j

Rt 0

Rt r ðsÞds 0 e : 1þC

r ðsÞds ¼ 1, then limt!1 NðtÞ ¼ K. The proof is complete.

h

Corollary 1.2. Let NðtÞ be a solution of Eq. (5) with Nð0Þ ¼ N 0 2 ð0; KÞ, then 0 < NðtÞ < K for all t 2 Rþ and the following results hold: Rt (i) limt!1 NðtÞ ¼ 0 if and only if limt!1 0 rðsÞds ¼ 1 holds. Rt (ii) limt!1 NðtÞ ¼ K if and only if limt!1 0 r ðsÞds ¼ 1 holds.

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Proof. Since N 0 2 ð0; KÞ, the solution of (5) can be rewritten as

NðtÞ ðK  NðtÞÞ1þC

¼

Rt r ðsÞds 0 e ; ðK  N0 Þ1þC N0

ð6Þ

which implies 0 < NðtÞ < K for all t 2 Rþ , and hence (i) and (ii) hold. The proof is complete.

h

Remark 1. From Corollary 1.2 and the formula (6), when NðtÞ converges to 0; limt!1 Rlnt NðtÞ  ¼ 1, that is to say C has nearly  0 rðsÞds Þ 1 no effect on the convergence rate. But when NðtÞ converges to K; limt!1 lnRðKNðtÞ ¼  1þC holds, which means that the convert 0

r ðsÞds

gence rate becomes slower and slower with the parameter C getting large.

1.2. Stochastic food-limited logistic population model The aim of this section is to study the effect of noise on stability of the equilibria. The main results about Eq. (4) are given as follows. Theorem 1.3. Let NðtÞ be a continuous positive solution to Eq. (4) for any initial value Nð0Þ ¼ N 0 with 0 < N 0 < K. h i (i) If there is a constant h > 0 such that rðtÞ  0:5r2 ðt Þ P h for all t P 0, then limt!1 E ðNðtÞ  K Þ2 ¼ 0. h i (ii) If there is a constant g > 0 such that r ðt Þ þ 0:5r2 ðt Þ 6 g for all t P 0, then limt!1 E ðNðtÞÞ2 ¼ 0. In particular, if rðtÞ 6 0:5r2 ðtÞ for all t P 0 and

h

i

r :¼ inf tP0 frðtÞg > 0, then limt!1 E ðNðtÞÞ3 ¼ 0.

By Theorem 1.3, we have the following result. Corollary 1.4. Let NðtÞ be a continuous positive solution to Eq. (4) for any initial value Nð0Þ ¼ N 0 with 0 < N 0 < K. Suppose that p > 0 is a constant.   (i) If there is a constant h > 0 such that rðtÞ  0:5r2 ðt Þ P h for all t P 0, then limt!1 E ðNðtÞ  K Þp ¼ 0. (ii) If either r ðt Þ þ 0:5r2 ðtÞ 6 g for some constant g > 0, or rðtÞ 6 0:5r2 ðtÞ and r :¼ inf tP0 frðtÞg > 0 hold, then limt!1 E ðNðtÞÞp ¼ 0. h i 2 Remark 2. Letting r ðt Þ  r and rðtÞ  r, from (i) of Theorem 1.3, we can get that r > r2 implies limt!1 E ðNðtÞ  K Þ2 ¼ 0. Thus Theorem 1.3 improves Theorem A obtained by Jiang et al. in [13]. Next, we present the results on the almost surely asymptotical stability of Eq. (4). Theorem 1.5. Let NðtÞ be a continuous positive solution toEq. (4) for any initial value Nð0Þ ¼ N 0 with 0 < N 0 < K. Denote 

Rt Rt 1 2 k ¼ lim inf t!1 1t 0 r ðsÞ  12 r2 ðsÞ ds and l ¼ lim supt!1 1t 0 rðsÞ þ 2ð1þC Þ r ðsÞ ds. (i) If k > 0, then the positive equilibrium K of (4) is almost surely asymptotically stable, i.e.,limt!1 N ðtÞ ¼ K, a.s.; (ii) If l < 0, then the zero solution of (4) is almost surely asymptotically stable, i.e., limt!1 N ðt Þ ¼ 0, a.s. By setting C  0 and applying Theorem 1.5 to (2), we have the result. Corollary 1.6. Let NðtÞ be a continuous positive solution to Eq. (2) for any

initial value Nð0Þ ¼ N 0 with 0 < N 0 < K. Denote Rt Rt k ¼ lim inf t!1 1t 0 r ðsÞ  12 r2 ðsÞ ds and l ¼ lim supt!1 1t 0 rðsÞ þ 12 r2 ðsÞ ds. (i) If k > 0, then limt!1 N ðt Þ ¼ K, a.s.; (ii) If l < 0, then limt!1 N ðtÞ ¼ 0, a.s. Remark 3. It is easy to verify that conditions of Corollary 1.6 are weaker than those of (A1)–(A2) presented in [1,4]. Thus Corollary 1.6 extends and improves the results (A1)–(A2). For example, if rðtÞ ¼ 0:1 þ 0:5 sin t and rðtÞ  0:2, then 2 k ¼ 0:08 > 0 and r ðt Þ  r 2ðtÞ ¼ 0:08 þ 0:5 sin t < 0 for t ¼ 2np þ 32p ; n 2 Z þ . That is to say, the condition of (A1) does not hold but the condition of (i) in Corollary 1.6 holds. If r ðt Þ ¼ 0:1 þ 0:5 sin t then l ¼ 0:08 < 0 and 2 r ðt Þ þ r 2ðtÞ ¼ 0:08 þ 0:5 sin t > 0 for t ¼ 2np þ p2 ; n 2 Z þ . In other words, the condition of (A2) does not hold but the condition of (ii) in Corollary 1.6 holds.

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Remark 4. Let r ðt Þ ¼ r and

r2 and rðtÞ  r be constants, then Theorem 1.5 shows the results under assumption: r <  2ðCþ1Þ

2

2

2

r r > r2 . A natural question is what happens if  2ðCþ1Þ < r < r2 , we answer that with the following discussion by using the Feller

test to (3) with constant coefficient

r > 0.

Theorem 1.7. Let NðtÞ be a continuous positive solution to Eq. (3) for any initial value Nð0Þ ¼ N 0 with 0 < N 0 < K. Suppose that r2 < r < r2 then  2ðCþ1Þ 2









  N0 bð1Þ  b ln KN 0



rC

P lim Nðt Þ ¼ 0 ¼ 1  P lim Nðt Þ ¼ K ¼ t!1

where bðxÞ ¼

Rx 0

2

e

Ru 0

bð1Þ  bð1Þ

t!1

bðv Þdv

du; bðxÞ ¼



r ðCþ1Þ

r2

þ 12

2 ð0; 1Þ;

1  r2 þ 1 1þe x satisfying 1 < bð1Þ 6 bð1Þ < 1.

Remark 5. Theorem 1.7 shows that it maybe impossible to find the threshold between persistence and extinction in almost sure sense of the population modeled by (3) and (4).

2. Proof of the main results We firstly prove a lemma to show that the solution will remain in ð0; KÞ with an initial value in the same interval, which is useful for the proof of our main results. Lemma 2.1. For any initial value 0 < N 0 < K, (4) has a uniquely uniformly continuous solution NðtÞ on t P 0 such that 0 < NðtÞ < K, a.s. Proof. Since the coefficients of the equation are local Lipschitz continuous, then for any initial value N 0 > 0, there exists a e e unique local solution NðtÞ on t 2 ½0;  s e Þ, where s e is the explosion time (cf. [17,18]). Define a stopping time se ¼ inf t 2 Rþ : xðt Þ R ð0; K Þ , then se 6 e s e Þ a.s. From the above, it’s true that for any 0 < N0 < K, there exist a unique local solution NðtÞ such that 0 < NðtÞ < K a.s. on t 2 ½0; se Þ. To show that the solution is global and 0 < N 0 < K implies that 0 < NðtÞ < K a.s. for all t 2 Rþ . We need only to show se ¼ 1 a:s. Let m0 2 Z þ be sufficiently large such that m10 6 x0 6 K  m10 . For any integer m > m0 , define the stopping time

sm ¼ inf t 2 ½0; se Þ : xðtÞ R



 1 1 ; ;K  m m

where throughout this paper we set inf ; ¼ 1 (as usual ; denotes the empty set). Clearly, sm is increasing as m ! 1. Set s1 ¼ limm!1 sm , then s1 6 se a:s. If we can show that s1 ¼ 1, then se ¼ 1 a:s. We assume that s1 ¼ 1 a:s. does not hold, then there have to be a pair of constants T > 0 and e 2 ð0; 1Þ such that Pðsm 6 T Þ P e. For all 0 6 t 6 sm ^ T, we define a C 2 function V : Rþ ! Rþ by

V ðNÞ ¼ 1  ln

N K N  ln ; K K

which is not negative for 0 < N < K. For m > m0 and 0 6 t 6 sm ^ T, by using Itô formula to (4) we get that

dV ðN ðt ÞÞ ¼ ¼

2NðtÞ  K N2 ðtÞ þ ðK  NðtÞÞ2 2 dN ðt Þ þ ðdNðt ÞÞ NðtÞðK  Nðt ÞÞ 2N 2 ðtÞðK  Nðt ÞÞ2 2NðtÞ  K N2 ðt Þ þ ðK  Nðt ÞÞ2 ½rðt Þdt þ rðt ÞdBðtÞ þ dt: K þ CNðtÞ 2ðK þ CN ðt ÞÞ2

For t 2 ½0; sm ^ TÞ, it’s clear that xðtÞ 2 ð0; KÞ holds. By integrating both sides from 0 to sm ^ T, and then taking expectations, it yields

EðV ðNðsm ^ T ÞÞÞ ¼ V ðNð0ÞÞ þ E

Z sm ^T ( 2NðtÞ  K 0

6 V ðN ð0ÞÞ þ E

K þ CNðtÞ

Z sm ^T 2NðtÞ K þ CNðtÞ 0

þ

N2 ðtÞ þ ðK  NðtÞÞ2

2ðK þ CNðtÞÞ2   þ 1 ds

6 V ðN ð0ÞÞ þ 3T: Note that for every x 2 fx : sm ðxÞ 6 T g , N ðsm ÞðxÞ ¼ m or

) ! ds

ð7Þ 1 , m

then

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D. Zhao, S. Yuan / Applied Mathematics and Computation 246 (2014) 599–607



EðV ðNðsm ^ T ÞÞÞ P E Ifsm 6T g V ðNðsm ÞÞ   1 K  m1 ¼ 1  ln m  ln Pðsm 6 T Þ K K   1 K  m1 e ! 1; P 1  ln m  ln K K as m ! 1, which contradicts with (7). Here and in the sequel, IA denotes an indicator function. Therefore, s1 ¼ 1 a:s. holds, and hence se ¼ 1 a:s. Now we prove the solution is uniformly continuous. From the above, the solution NðtÞ of Eq. (4) satisfies 0 < NðtÞ < K. By NðtÞðKNðtÞÞ the continuousness and boundedness of rðtÞ and rðtÞ, we can see that f ðxðtÞ; tÞ ¼ rðtÞ NðtÞðKNðtÞÞ ðKþCNðtÞ and gðxðtÞ; tÞ ¼ rðtÞ ðKþCNðtÞ

satisfy the local Lipschitz condition and the linear growth condition. EðN p ðtÞÞ 6 K p < 1 holds for every p > 0. Then by Mao [19, Lemma 2.4], almost every sample path of NðtÞ is uniformly continuous on t > 0. The proof is complete. h Lemma 2.2 See [13]. Let f be a nonnegative function defined on ½0; 1Þ such that f is integrable on ½0; 1Þ and is uniformly continuous on ½0; 1Þ. Then limt!1 f ðt Þ ¼ 0. Proof of Theorem 1.3. Let NðtÞ be a solution with initial value 0 < N 0 < K. It follows from Lemma 2.1 that 0 < NðtÞ < K on t P 0. We define a Lyapunov function VðtÞ by

VðtÞ ¼

NðtÞ NðtÞ  1  ln ; K K

t P 0:

By Itô’s formula, we have

" # 1 1 1 2 dVðtÞ ¼ dNðtÞ  dNðtÞ  ðdNðtÞÞ K NðtÞ 2N2 ðtÞ 

   1 1 NðtÞ½K  NðtÞ r2 ðtÞ K  NðtÞ 2  ½r ðt Þdt þ rðt ÞdBðt Þ þ dt K NðtÞ K þ CNðtÞ 2 K þ CNðtÞ

¼

ð8Þ

  ðK  NðtÞÞ2 r2 ðtÞ K  NðtÞ 2 ½r ðt Þdt þ rðt ÞdBðtÞ þ dt ¼ 2 K þ CNðtÞ K ðK þ CNðtÞÞ    ðK  NðtÞÞ2 r2 ð t Þ r2 ðtÞ CNðtÞðK  NðtÞÞ2 dt þ rðt ÞdBðt Þ  r ðt Þ  dt: ¼ 2 2 K ðK þ CNðtÞÞ K ðK þ CNðtÞÞ2 Integrating both sides of equality (8), we get

VðtÞ þ

Z

t

0

  Z t Z t ðK  NðsÞÞ2 r2 ð sÞ ðK  NðsÞÞ2 ds ¼ Vð0Þ  r ðsÞ  rðsÞdBðsÞ  2 K ðK þ CNðsÞÞ 0 K ðK þ CNðsÞÞ 0

r2 ðsÞ CNðsÞðK  NðsÞÞ2 2

K ðK þ CNðsÞÞ2

Taking the expectation of both sides of equality (9), we obtain

E½VðtÞ þ E

"Z

t

0

"Z   # t ðK  NðsÞÞ2 r2 ð sÞ ds ¼ E½Vð0Þ  E r ðsÞ  2 K ðK þ CNðsÞÞ 0

r2 ðsÞ CNðsÞðK  NðsÞÞ2 2

K ðK þ CNðsÞÞ2 2

# ds :

Noting that 0 < N ðtÞ < K implying V ðt Þ P 0 for all t P 0, by assumption r ðt Þ  r 2ðtÞ P h we have that

E½VðtÞ þ

h 2

K ð1 þ C Þ

Z

t

h i E ðK  NðsÞÞ2 ds 6 E½Vð0Þ < 1;

0

h i which implies E ðNðtÞ  K Þ2 2 L1 ½0; 1Þ. By Lemma 2.2 one obtains

h i lim E ðNðtÞ  K Þ2 ¼ 0:

t!1

Thus we complete the proof of (i) in Theorem 1.3. Similar to proof of (i), We define a Lyapunov function VðtÞ by

VðtÞ ¼

K  NðtÞ K  NðtÞ  1  ln ; K K

t P 0:

ds:

ð9Þ

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D. Zhao, S. Yuan / Applied Mathematics and Computation 246 (2014) 599–607

By Ito’s formula, we have

dVðtÞ ¼

N2 ðtÞ K ðK þ CNðtÞÞ

   r2 ðtÞ r2 ðtÞ CN3 ðtÞ dt þ rðt ÞdBðtÞ  r ðt Þ þ dt: 2 2 K ðK þ CNðtÞÞ2

ð10Þ

Integrating both sides of equality (10) and then taking the expectation, we have

E½VðtÞ  E

"Z

"Z   # t N2 ðsÞ r2 ð sÞ ds ¼ E½Vð0Þ  E r ðsÞ þ K ðK þ CNðsÞÞ 2 0

t

0

r2 ðsÞ

CN3 ðsÞ

2

K ðK þ CNðsÞÞ2

# ds :

2

Noting that 0 < N ðtÞ < K implying V ðtÞ P 0 for all t P 0, by assumption r ðt Þ þ r 2ðtÞ 6 g we have

E½VðtÞ þ

Z

g 2

K ð1 þ C Þ

t

h i E N2 ðsÞ ds 6 E½Vð0Þ < 1;

0

h i which implies that E N 2 ðtÞ 2 L1 ½0; 1Þ. By Lemma 2.2 one obtains

h i lim E N 2 ðtÞ ¼ 0:

t!1

h i By using (10) again, for r ðt Þ 6 0:5r2 ðt Þ for all t P 0 and r :¼ inf tP0 frðt Þg > 0, we get limt!1 E ðNðtÞÞ3 ¼ 0. Thus we complete the proof.

Proof of Corollary 1.4. For any constant p P 2, by use of the fact 0 6 NðtÞ 6 K, we compute that

 p  h i   K  NðtÞ 6 K p2 E ðK  NðtÞÞ2 : E ðK  NðtÞÞp ¼ K p E K When 0 < p < 2, by the moment inequality, we get

i2p    h E ðK  NðtÞÞp 6 E ðK  NðtÞÞ2 :   An application of Theorem 1.3 gives that limt!1 E ðK  NðtÞÞp ¼ 0 for any p > 0. Similarly, one can prove that p limt!1 E½N ðtÞ ¼ 0. Proof of Theorem 1.5. By Itô’s formula, we have from (4)

d ln

"

NðtÞ ½K  NðtÞCþ1

# " # 1 1 1 1 2 2 dNðtÞ  ¼ dNðtÞ  ðdNðtÞÞ  ðC þ 1Þ ðdNðtÞÞ : NðtÞ NðtÞ  K 2N 2 ðtÞ 2½NðtÞ  K 2

ð11Þ

Substituting Eq. (4) into equality (11), then

d ln

NðtÞ ½K  NðtÞ

¼

Cþ1

K þ CNðtÞ 1 ½NðtÞ  K 2  ðC þ 1ÞN2 ðtÞ 2 dNðtÞ  ðdNðtÞÞ NðtÞðK  NðtÞÞ 2 N2 ðtÞ½NðtÞ  K 2

¼ ðr ðt Þdt þ rðt ÞdBðtÞÞ  ¼

r2 ðtÞ K 2  2KNðtÞ  CN2 ðtÞ 2

ðK þ CNðtÞÞ2

ð12Þ

dt

   r2 ð t Þ r2 ðtÞ NðtÞð2K þ CNðtÞÞ r ðt Þ  dt þ rðt ÞdBðt Þ þ ðC þ 1Þ dt: 2 2 ðK þ CNðtÞÞ2

Integrating both sides of the equality (12), we have

ln

NðtÞ ½K  NðtÞ

where M ðt Þ ¼

Rt 0

Cþ1

¼ ln

N0 ½K  N 0 

Z

t

r2 ðsÞ NðsÞð2K þ CNðsÞÞ 2

ðK þ CNðsÞÞ2

ds þ MðtÞ;

ð13Þ

Noting the boundedness of

M ðt Þ ¼ 0; t





r2 ðsÞds 6 t  sup r2 ðsÞ :

0

lim

 Z t Z t r2 ðsÞ ds þ ðC þ 1Þ r ðsÞ  2 0 0

rðsÞdBðsÞ is a martingale with quadratic variation

hM ðtÞ; M ðt Þi ¼

t!1

þ Cþ1

sP0

rðtÞ, by the strong law of large numbers for martingales, we can see that

a:s:

In view of the definition of k and k > 0, it follows that for arbitrary e > 0, there exists T > 0 such that for t > T, the following inequality holds.

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D. Zhao, S. Yuan / Applied Mathematics and Computation 246 (2014) 599–607

 Z t Z t r2 ðsÞ ds þ r ð sÞ  rðsÞdBðsÞ P ðk  eÞt: 2 0 0

ð14Þ

Noting that 0 < N ðtÞ < K for all t P 0, by (13) and (14) we have

ln

NðtÞ ½K  NðtÞCþ1

P ln

N0 ½K  N0 Cþ1

þ ðk  eÞt;

a:s:

By choosing e be sufficiently small such that k  e > 0, it then follows from 9 that limt!1 N ðt Þ ¼ K; a:s. Thus we complete the proof of (i) in Theorem 1.5. Next, we prove (ii) of Theorem 1.5. We rewrite (12) to be

d ln



NðtÞ ½K  NðtÞCþ1

¼

r ðt Þ þ

  r2 ðtÞ ððC þ 2ÞK þ CNðtÞÞðK  NðtÞÞ dt þ rðt ÞdBðt Þ  dt: 2ðC þ 1Þ 2 ð1 þ CÞðK þ CNðtÞÞ2

r2 ð t Þ

ð15Þ

Integrating both sides of equality (15), by calculation we have that for t large enough

ln

NðtÞ ½K  NðtÞCþ1

¼ ln

þ Cþ1

Z t r ðsÞ þ

r2 ðsÞ

 ds

2ð C þ 1Þ ½K  N0  0 Z t 2 r ðsÞ ððC þ 2ÞK þ CNðsÞÞðK  NðsÞÞ ds þ MðtÞ  2 ðC þ 1ÞðK þ CNðsÞÞ2 0 6 ln

x(t)

N0

N0 ½K  N0 Cþ1

 ðl  eÞt:

1

1

0.8

0.8

0.6

x(t)

0.6

0.4

0.4

0.2

0.2

0

0

100

200

ð16Þ

300

400

0

500

0

100

(a) Time

x(t)

1

0.8

0.8

0.6

x(t)

0.4

0.2

0.2

100

200

300

(c) Time

400

500

400

500

0.6

0.4

0

300

(b) Time

1

0

200

400

500

0

0

100

200

300

(d) Time

Fig. 1. The lines represent the solution of (4) for rðtÞ ¼ 0:05 þ 0:5sint; K ¼ 1, initial data N 0 ¼ 0:4, step size Dt ¼ 0:001 with rðtÞ ¼ 0:3; C ¼ 0 in (b), rðtÞ ¼ 0; C ¼ 1:2 in (c) and rðtÞ ¼ 0:3; C ¼ 1:2 in (d).

rðtÞ ¼ 0; C ¼ 0 in (a),

606

D. Zhao, S. Yuan / Applied Mathematics and Computation 246 (2014) 599–607

By choosing ebe sufficiently small such that l  e > 0, it then follows from (16) that limt!1 N ðtÞ ¼ 0; a:s. The proof is complete. Proof of Theorem 1.7. By using Itô’s formula to (3), we have

" # " # NðtÞ 1 1 1 1 2 2 d ln dNðtÞ  ðdNðtÞÞ  ðdNðtÞÞ ¼ dNðtÞ  K  NðtÞ NðtÞ NðtÞ  K 2N2 ðtÞ 2½NðtÞ  K 2   K r2 ðK  2NðtÞÞ ¼ rdt þ rðtÞdBðtÞ  dt : K þ CNðtÞ 2ðK þ CNðtÞÞ

ð17Þ

NðtÞ Denote xðtÞ ¼ ln KNðtÞ , then

dxðt Þ ¼ uðxðt ÞÞ½/ðxðt ÞÞdt þ rdBðt Þ; xðt Þ

2

where uðxðtÞÞ ¼ 1þð1þe and /ðxðtÞÞ ¼ r  r2 Cþ1ÞexðtÞ

bðxÞ ¼

/ðxÞ

r2 uðxÞ

¼

 rðC þ 1Þ

r2

1exðtÞ . 1þðCþ1ÞexðtÞ

Let bðxÞ ¼

Rx

   1 rC 1 þ þ 1 ;  2 1 þ ex r2

0

2

e

Ru 0

bðv Þdv

du with

then we get that

2

Z

x



bðv Þdv ¼ ½2rðC þ 1Þ þ r2 x þ 2 rC þ r2 ln

0

ex  ½2rðC þ 1Þ þ r2 x; 1 þ ex

when x ! 1; and

2

Z

x

bðv Þdv  ½2rðC þ 1Þ þ r2  2rC  2r2 x ¼ ½2r  r2 x;

0 2

2

r when x ! 1. Therefore, for  2ðCþ1Þ < r < r2 , we have

1 < bð1Þ 6 0 6 bð1Þ < 1: Thus an application of the Feller test [6, Proposition 2.1] gives the desired result. The proof is complete.

3. Simulation and discussion In this section we shall work out some simulations to illustrate the analytical results by employing the Euler–Maruyama (EM) method (see [20]). Eq. (4) takes the form

X jþ1 ¼ X j þ r ðjDt Þ





Xj K  Xj Xj K  Xj Dt þ rðjDt Þ ½Bððj þ 1ÞDtÞ  BðjDt Þ; K þ CX j K þ CX j

j ¼ 0; 1; 2; . . . ; L;

where X j ¼ N ðjDt Þ; Bððj þ 1ÞDtÞ  BðjDtÞ is an increments generated by using discretized Brownian paths like that in [20].

x(t)

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

x(t)

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

0

50

100

150

(e) Time

200

250

300

0

50

100

150

200

250

300

(f) Time

Fig. 2. The lines represent the solutions of (4) for rðtÞ ¼ 0:1 þ 0:2sint; K ¼ 1; C ¼ 2, initial data N 0 ¼ 0:4, step size Dt ¼ 0:001 with rðtÞ ¼ 0:6 in (f).

rðtÞ ¼ 0 in (e) and

D. Zhao, S. Yuan / Applied Mathematics and Computation 246 (2014) 599–607

607

In Fig. 1, (a) and (c) are deterministic trajectories of (5) with the only difference on the value of C. (b) and (d) are stochastic trajectories of the stochastic model (4) with the only difference of C. By Theorem 1.1, the solution of (5) converges to the positive equilibrium K, see (a) and (c). In model (4), we choose rðtÞ ¼ 0:05 þ 0:5sint; K ¼ 1; rðtÞ  0:3 and C ¼ 0 or C ¼ 1:2. Compute that k ¼ 0:005 > 0, by use of (i) in Theorem 1.5, the positive equilibrium Kis almost surely asymptotically stable, which does not depend on the parameter C. Fig. 1(b) and Fig. 1(d) confirm this. In Fig. 2, (e) represents deterministic trajectory of (5) for rðtÞ ¼ 0:1 þ 0:2sint; K ¼ 1; C ¼ 2 and initial data N 0 ¼ 0:4. (b) represents stochastic trajectory of (4) for rðtÞ ¼ 0:1 þ 0:2sint; K ¼ 1; C ¼ 2; rðtÞ ¼ 0:6 and initial data N 0 ¼ 0:4. Corollary 1.2 shows the solution of (5) converges to zero, see Fig. 2(e). Theorem 1.5 ensures that the equilibrium 0 is almost surely asymptotically stable since l ¼ 0:04 < 0. Fig. 2(f) confirms this. In this paper, we establish some criteria for the asymptotic behavior of the Eq. (4). Sufficient conditions are obtained for the solution converging to the equilibria: 0 and K, which improve Theorem A obtained by Jiang et al. in [13]. Criteria are also given to make sure the positive solution converging almost surely to the equilibria. They extend and improve previously known results presented by Golec and Sathananthan in [4]. Obtained Results indicate that the food-limited assumption has an effect on the convergence rate of the solution, which are supported by the numerical simulations. Comparing the trajectory in Fig. 1(a) and (c), one can see the parameter C has an influence on the convergence rate when the population converges to the positive equilibrium as stated in Remark 1. At the same time, when we compare the trajectory in Fig. 1(b) and (d), the same phenomenon occurs. It is therefore interesting to find out how the food-limited assumption affects the persistence and extinction of the population modeled by the stochastic Eq. (4). Theorem 1.5 answers only a little part of these phenomena. We leave the left for future. Acknowledgements This work was supported by NSFC (No. 11271260), Training Program for NSFC of USST (No. 14XPQ03), The Hujiang Foundation of China (B14005) and Shanghai Leading Academic Discipline Project (No. XTKX2012). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.amc.2014.08.070. References [1] D. Jiang, X. Zhang, D. Wang, N. Shi, Existence, uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation, Sci. China Ser. A 50 (2007) 977–986. [2] X. Sun, Y. Wang, Stability analysis of a stochastic logistic model with nonlinear diffusion term, Appl. Math. Model. 32 (2008) 2067–2075. [3] M. Liu, K. Wang, Stability of a stochastic logistic model with distributed delay, Math. Comput. Model. 57 (5–6) (2013) 1112–1121. [4] J. Golec, S. Sathananthan, Stability analysis of a stochastic logistic model, Math. Comput. Model. 38 (2003) 585–593. [5] M. Krstic, M. Jovanovic, On stochastic population model with the Allee effect, Math. Comput. Model. 52 (2010) 370–379. [6] Q. Yang, D. Jiang, A note on asymptotic behaviors of stochastic population model with Allee effect, Appl. Math. Model. 35 (2011) 4611–4619. [7] F.E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology 44 (1963) 651–663. [8] K. Gopalsamy, M.R.S. Kulenovic, G. Ladas, Environmental periodicity and time delays in a food-limited population model, J. Math. Anal. Appl. 147 (1990) 225–237. [9] S. Tang, L. 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Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164–172. [16] F. Chen, Some new results on the permanence and extinction of nonautonomous Gilpin–Ayala type competition model with delays, Nonlinear Anal.: RWA 7 (2006) 1205–1222. [17] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972. [18] A. Freedman, Stochastic Differential Equations and Their Applications, vol. 2, Academic Press, San Diego, 1976. [19] X. Mao, Stochastic versions of the Lassalle theorem, J. Differ. Equ. 153 (1999) 175–195. [20] D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001) 525–546.